Abelian variety isogeny class 4.2.ab_a_a_d over $\F_{2}$

• Label: 4.2.ab_a_a_d
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$1 - x + 3 x^{4} - 8 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.142014315702$, $\pm0.271261639785$, $\pm0.607066675494$, $\pm0.840353784243$
Angle rank:	$4$ (numerical)
Number field:	8.0.2917030625.1
Galois group:	$C_2 \wr C_2\wr C_2$
Jacobians:	$1$
Cyclic group of points:	yes

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11$	$319$	$4169$	$143231$	$1600951$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$4$	$8$	$28$	$47$	$82$	$93$	$308$	$530$	$999$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $x^2+x y+y^2+z t=x^2 y+y^3+x^2 z+y^2 z+y z^2+x^2 t+x t^2=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 8.0.2917030625.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
4.2.b_a_a_d	$2$	4.4.ab_g_a_z